Please tell me how or what formula is used when calculating bracelet sizes? I have been estimating bracelet sizes for years and very weary of this method. Is there a mathematical way of calculating?
Yes, there is. In mathematical lingo, to find the size of a bracelet
is to find circumference of an ellipse. In case of circle the task is
trivial, but ellipse is unexpectedly difficult. The method of taking
length plus width, dividing it by 2 and using result as diameter of a
circle, - works in creating oval settings. In case of bracelet, this
method will be off by few millimeters. So better way is needed.
There are 2 ways to go, - to use approximation which is good enough
for our purposes, or to calculate precisely. I am not going to
explain precise method, because it involves computing infinite
series. Precision gained is meaningless for working with metal. It
will be a lot of work with very small yield. I am not going to
mention it here.
Very good approximation is given by Ramanujan. Let C be
circumference; a - one half of ellipse length and b - one half of
ellipse width. The formula then becomes -
C = 3.14159*[ 3a + 3b - sqrt( (3a + b) * (a + 3b) ) ]
Let’s check it out. Ellipse 60 mm long and 40 mm wide. Computing by
circle method = 3.14159*((60 + 40)/2) = 157mm. Let’s try Ramanujan
way -
C = 3.14159 * [ 330 + 320 - sqrt( (330 + 20) * (30 + 320))]
simplifying - 3.14159 * [ 150 - sqrt( 110 * 90 ))] = 3.14159 * ( 150
- sqrt(9900)) = 3.14159 * ( 150 - 99.5) = 3.14159 * 50.5 = 158.7
The difference is 1.7mm. Large dimensions would produce larger
errors.
Leonid Surpin